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10 SPACES WITH OPERATORS. ORBIT SPACES 51
x e E and all, s, t in G. Everything we have said can be immediately transposed
to this situation; the set of orbits is sometimes denoted by G\E. For example,
a subgroup H of a topological group G acts continuously on the right on G by
the action (s, x)\~+xs.
1. Let G be a locally compact metrizable group and E a locally compact metrizable space
on which G acts continuously. For each pair of subsets K, L in E let P(K, L) denote
the set of all s e G such that (s • K) n L ^ 0.
(a) Show that if K is compact and L is closed in G, then the set P(K, L) is closed
in G. The group G is said to act properly on E if P(K, L) is a compact subset of G
whenever K and L are compact subsets of E. (This will always be the case if G is
(b) Show that, if G acts properly on E, then F • K is closed in E whenever F is closed
in G and K is a compact subset of E. In particular, for each x e E, the orbit G • x is
closed in E.
(c) Under the same hypotheses, for each x e E the stabilizer S* of x is a compact
subgroup of G, and the canonical map G/S* -> G • x is a homeomorphism.
(d) Under the same hypotheses, the orbit space E/G is Hausdorff.
2. For each pair (a, t) of real numbers such that a^l, the point fa(t) eR2 is defined as
Let Cfl denote the set of points fa(t) (t e R) and let E be the union of the sets Ca (a ^ 1)
and the lines D', D*, where D' (resp. D*) is the set of points (/, —1) (resp. (t, 1)) with
The additive group R acts on the locally compact space E as follows:
(1) s - (t, -1) = (s 4- 1, -1); (2) 5 •/«(/) =/0 + /) for a £ 1 ; (3) s - (t, 1) * (/- j, 1).
Show that R acts continuously and freely on E, that the orbits R • z are closed sets and
that the canonical mapping R~>R • z is a homeomorphism, but that the orbit space
E/R is not Hausdorff.
3. The group Z acts continuously and properly on R2 as follows :
The orbit space M (the Mobius strip) is metrizable and locally compact. If TT : R2 -»• M
is the canonical mapping, the restriction of TT to E — D" (notation of Problem 2) is
injective, and ?r(E — DO = 7r(E) is locally compact. The group R acts on u(E — DO
by the law s - rr(z) = TT($ • z) for all z e E — D*. Show that R acts continuously andgroup G on a set E is a mapping (s, x)\-*x- s